142 FROM NEBULA TO NEBULA 



And what are the laws of falling bodies? These are 

 given by Ganot (Art. 49) in these words : 



1. The velocities are proportional to the times dur- 

 ing which the motion has lasted. 



2. The spaces described are proportional to the 

 squares of the times employed in their description. 



3. The spaces described are proportional to the 

 squares of the velocities acquired during their descrip- 

 tion. 



4. The spaces described in equal successive periods 

 of time increase by a constant quantity. 



Suppose a bar, whose longer arm we shall call R and 

 its shorter r, be rotated horizontally around a stationary 

 pivot, it will then descibe two circles, and, by geometry, 

 we get the equation, 



r :R ::2irr :2-*R (1) 



By our hypothesis, however, the pivot is not station- 

 ary, but is falling, and the ends of the bar are not describ- 

 ing closed circles in fact, but coils of open spirals. In 

 still other words, the last two terms of our equation are, 

 properly construed, "spaces fallen through." Under 

 the second rule given above, then, our last two terms, in 

 order to express the element of time instead of space, 

 must be amended to read 



V 2*r : V 2>*R (2) 



At this point, let us not forget that we are not dealing 

 with a simple bar, but with a lever, whose arms, in order 

 that they may balance, must be weighted inversely as 

 their length; that is to say, a weight R must be attached 

 to the end of arm r, and a weight r to the end of arm R. 

 Pound for pound, then, the time required to complete the 

 running of the smaller coil as compared with that re- 

 quired for the larger is : 



~ ^ -\/^ W (3) whence, (parts of circles being 



7T 7 V^TTjn . -iji j_ i 



- . to each other as their 



like parts) 



r^/~r~:R\/TT (4) meaning, 



